Within the Floquet–Bloch–Gelfand theory, see e.g. [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, mathematical studies of
spectra with the band-gap structures need to find out the eigenvalues
of spectral elliptic boundary value problems which are posed in the
periodicity cell and involve an additional continuous parameter
\(\eta \in [-\pi ,\pi ]\) , the Floquet parameter or the Gelfand dual variable.
It is a very rare situation when such a problem admits
explicit solutions while computational methods become rather
expensive to present the whole family of dispersion curves,
projections of which on the ordinate \(\lambda \) -axis involve the
spectral bands. As usual, variational and asymptotic methods help to
prove or disprove the existence of open spectral gaps in a certain
range of the spectrum and to estimate their geometrical
characteristics.
